In conjectural variation models
the firms in the industry are taken as given and each firm makes certain
assumptions about what the others reactions will be to its own actions.
For example, in the Cournot model each firm assumes there will be no
reaction on the part of the other firms. In the limit price models
one firm
chooses its action taking into account the possible entry or exit of
competitors to or from the market.

Conjectural Variation Models of Oligopoly

Undifferentiated Product Market

Suppose there are n firms in the market and the inverse demand function
for the market is:

(1) p = p0 - b*Q,

where Q is the total production of all the firms in the market.
Let the cost function for the i-th firm be

(2) Ci = Ci0 + Ci1*qi,

where qi is the output of the i-th firm. The profit of the
i-th firm, Ui, is then

(3) Ui = p*qi - Ci = (p0 - b*Q)*qi - Ci0 - Ci1*qi.

The first order condition for maximizing Ui with respect
to qi is:

(4) ∂Ui/∂qi
=
(p0 - b*Q) -b*(∂Q/∂qi)*qi
- Ci1 = 0.

The Cournot Model

In the Cournot Model each firm presumes no reaction on the part of
the other firms to a change in its output. Thus,
∂Q/∂qi = 1.
Therefore the first order condition for a maximum profit of the i-th
firm is:

(5) p0 - b*(Qoi + 2qi)
= Ci1,

where Qoi is the output of the firms other than the i-th.
When this is solved for qi the result is:

(6) qi = (p0 - Ci1)/2b
- Qoi/2.

However it is more convenient to represent the first order condition and
its solution as:

(7) p0 - b*(Q + qi) = Ci1
and qi = (p0 - Ci1)/b - Q.

Now we can sum the above equation over the n firms. The result is:

(8) Q = n(p0/b) - C1/b - n*Q,

where C1 is the sum of the Ci1. The solution for Q is:

(9) Q = [n/(n+1)](p0/b) - [1/(n+1)]C1/b.

When this output is substituted into the inverse demand function the result
is:

(10) p = [1/(n+1)]p0 + [1/(n+1)]C1,

or if we let c1=C1/n:

(11) p = [1/(n+1)]p0 + [n/(n+1)]c1,

where c1 represents the average of the marginal costs of the
n firms.
We see from (11) that as the number of firms increase without
bound the
market price approaches c1.

If one follows through with this model one would have to take in consideration that
the firms with above average
marginal cost could be making a loss on variable costs and would cease
production.

The von Stackelberg Leader-Follower Model

Heinrich von Stackelberg proposed a model of oligopoly in which one firm, a
follower,
takes the output of the other firm as given (a Cournot type oligopolist) and
adjusts its output accordingly. The other firm, a leader, takes into account
the adjustment which the follower firm will make. The output decision of
a Cournot oligopolist is given by equation (6) above. Thus if a leader firm
increases its output qL by 1 unit the follower firm will decrease its output
by one half of a unit. The term ∂Q/∂qL = 1/2 for the
leader firm so the first order condition for the leader firm is:

(12) ∂UL/∂qL
= (p0 - b*Q) -b*(1/2)*qL - CL1 = 0.

Thus

(13) qL = (p0 - CL1)(2/3b)
- 2QoL/3.

Carrying through with the analysis as shown below indicates that the market
price will be:

(14) p = [1/(n+2)]p0 + [(n+1)/(n+2)]c1,

where c1 is now the weighted average of the marginal costs of
the firm with all of the follower firms given an equal weight and the
leader firm given a weight of twice that of the follower firms. The
leader
firm has the effect on the industry of two follower firms. Otherwise the
result is the same as in the case of the Cournot oligopoly.

The General Case

(16) Q = N(p0/b) - Nc1/b - NQ,

where N is the sum of the weights Wi and c1 is the weighted
average of the marginal costs Ci1. Thus,

(17) Q = [n/(n+1)]*(p0-c1)/b.

This result when substituted into the inverse demand function gives:

(18) p = [1/(n+1)]p0 + [n/(n+1)]c1.

This is the same as the Cournot solution with the number of firms
replaced
by the effective number of competitors, the sum of the reciprocals of
(∂Q/∂qi). From (18) we also have that the change in
oligopoly price is a weighted average of the shifts in the marginal costs
and shifts in the demand function as given by the parameter p0.

Differentiated Products

For this case it is convenient and also necessary to use a matrix
formulation
of the problem. It is assumed that each firm produces a different
product.
Let P and Q be the column vectors of prices and outputs. The inverse demand
function is taken to be linear and of the form:

(19) P = P0 - BQ.

The cost function for each firm is of the form Ci
= C0i + C1i*qi. The first
order condition for a maximum profit with respect to qi is:

(20) pi0 - Σj[bij*qj]
+ qi*[-Σj[bij*(∂qj/∂qi)]
- Ci1 = 0,

where Σj[] denotes the summation with respect to the index j. The set of
these first order conditions for i=1,...,n can be represented in matrix form
as:

(21)P - BQ - DQ - C1 = 0,

where D is a diagonal matrix whose diagonal elements are:

(22) dii = Σj[bij*(∂qj
/∂qi).

It is the diagonal matrix created from the diagonal elements of
the matrix BJ, where J = [(∂qj/∂qi)].

The solution for Q is

(23) Q = G(P0 - C1),

where G is the inverse of (B+D). The vector of market prices is:

(24) P = (I-BG)P0 + BGC1.

Thus the oligopoly prices given as P are weighted averages of the P0
parameters and the marginal costs C1.

Limit Pricing Models of Oligopoly

An abreviated version of a limit pricing model of oligopoly is given at
Limit.